Question

Find ﻿f+gf+gf+g﻿, ﻿f−gf-gf−g﻿, ﻿fgfgfg﻿, and ﻿fg\frac{f}{g}gf​﻿. Determine the domain for each function.﻿f(x)=6x2−x−1f\left(x\right)=$$6x^2-x-1f(x)=6x2$$−x−1﻿, ﻿g(x)=x−1g\left(x\right)=x-1g(x)=x−1

Accepted Answer

Step 1: Understand the problem. You are tasked with finding the domain of the functions f(x) = $$6x^2 - x - 1 $$and g(x) = x - 1, as well as the domain of their combinations: f + g, f - g, fg, and f/g. The domain of a function is the set of all x-values for which the function is defined. Step 2: Analyze the domain of f(x). Since f(x) = $$6x^2 - x - 1 is a $$polynomial, it is defined for all real numbers. Therefore, the domain of f(x) is all real numbers. Step 3: Analyze the domain of g(x). The function g(x) = x - 1 is also a polynomial, so it is defined for all real numbers. Thus, the domain of g(x) is all real numbers. Step 4: Determine the domain of f + g, f - g, and fg. Since these operations involve adding, subtracting, or multiplying two polynomials, the resulting functions are also polynomials. Therefore, the domain of f + g, f - g, and fg is all real numbers. Step 5: Determine the domain of f/g. Division by zero is undefined, so we must exclude any x-values that make g(x) = 0. Solve g(x) = x - 1 = 0, which gives x = 1. Therefore, the domain of f/g is all real numbers except x = 1.

Find $f+g$ , $f-g$ , $fg$ , and $\frac{f}{g}$ . Determine the domain for each function.
$f\(\left\)(x\(\right\))=6x^2-x-1$ , $g\(\left\)(x\(\right\))=x-1$

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Key Concepts

Function Operations

Domain of a Function

Rational Functions

Find f+gf+gf+g, f−gf-gf−g, fgfgfg, and fg\(\frac{f}{g}\)gf. Determine the domain for each function.f(x)=6x2−x−1f\(\left\)(x\(\right\))=6x^2-x-1f(x)=6x2−x−1, g(x)=x−1g\(\left\)(x\(\right\))=x-1g(x)=x−1